Abstract:
The Cosmic Microwave Background (CMB) is the oldest light in the Universe and gives
us access to a picture of the Universe when it was only ≈ 300,000 years old. The CMB
radiation consists of photons free streaming to us from the surface of last scattering. The
existence of such radiation was predicted by Dicke and his group and was experimen-
tally detected by Penzias and Wilson in 1965. The CMB gives us a window to study the
fundmental laws of physics by observing the Universe on the largest scales. Precise mea-
surements of the temperature and polarization fields of the CMB have allowed us to gain
valuable physical insights about properties of our Universe. It is found that on length scales
larger than ≈ 300 Mpc, our Universe is homogeneous and statistically isotropic. In the first
part of our work, we test the fundamental assumption of SI and in the second part, we test
the prediction of noncommutative spacetime.
The standard ΛCDM cosmological model is the most widely accpeted model of our
Universe. The assumption of Statistical Isotropy (SI) is one of the important assumptions
of the ΛCDM model and has been found to be consistent with most tests of isotropy using
observations. However, there are many models which predict that our Universe is not
isotropic and strong limits on the violation of SI can constrain these models. There isn’t
any a priori reason for the Universe to be statistically isotropic, and hence it is important
to test this fundamental assumption which is one of the pillars of our understanding of the
Universe. We have developed techniques to measure the isotropy of random fields on the
sphere and in three dimensional flat space, and have used cosmological data to search for
deviations from SI.
We use Minkowski Tensors (MTs), which carry information regarding the shape of
closed curves, as a measure of isotropy. By choosing a suitable threshold to cut off a given
field, we get closed curves which form the boundaries of the connected regions and holes.
The isotropy and alignment of these closed curves provide information on the SI of the
underlying field. MTs were previously defined for closed curves in flat spaces. However,
cosmological fields such as the CMB are defined on the sphere and thus the MTs can not
be directly applied to these fields. We generalize the definition of MTs to closed curves
on the sphere and provide a numerical method to estimate the MTs from pixelated maps.
Further, we apply our technique to the CMB temperature data given by the Planck mission
and find no signficant deviation from SI. We also apply our method to the beam convolved
individual frequency CMB temperature maps given by Planck and find that they are all
consistent with SI, except for the 30 GHz maps, which exhibit a mild level of anisotropy.
We suspect that an inaccurate estimation of the instrument beam or residual noise at 30
GHz could be the primary reason for this mild discrepancy.
We have also used the MTs for three dimensional density fields to demonstrate a method
to constrain the effect of Redshift Space Distortion (RSD). RSD is the modification of
the apparent shape of galaxy clusters due to the peculiar velocity of the galaxies, which
affects the measured redshift of the galaxy, making the Hubble’s redshift-distance relation
an inaccurate approximation. We compute the ensemble expectation values of the MTs
for a 3D isotropic Gaussian field and develop a numerical and a semi-analytic method to
estimate the MTs from a discretely sampled field. We apply our method to estimate the
MTs from the isotropic fields and the anisotropic fields obtained by applying a linear RSD
operator to the isotropic feilds. We find that RSD leads to a shift in the amplitude of the
elements of the MTs and show that this shift can be used to obtain constraints on the linear
RSD parameter.
In the last part of our work, we have tested spacetime noncommutativity, which is an
essential prediction of string theory. Based on evidence from observations, it is believed
that the early Universe went through a phase of accelerated expansion, known as Inflation.
During this phase, very small regions of space were blown up to cosmological sizes in
a very short amount of time. This allows us to search for spacetime noncommutativity,
which is an effect relevant at the extremely small scales, by looking at cosmological fields
such as the CMB. If our spacetime is noncommutative, then the form of the CMB angular
power spectrum is modified and this effect can be used to constrain the energy scale of
spacetime noncommutativity. First we estimate the effect of noncommutative spacetime on
the CMB angular power spectrum using the publicly available package, CAMB, and show
that Planck data is best suited to constrain the spacetime noncommutativity parameter. We
then perform a Bayesian analysis of the Planck 2013 CMB temperature data and obtain a
lower bound of ≈ 20 TeV on the energy scale of spacetime noncommutativity. Our results,
improve upon those of previous works by a factor of two.
Finally we summarize all the research work that was carried out as part of this thesis.
We also discuss prospects for further research in the context of future CMB experiments
having higher resolutions than Planck and the application of MTs for probing the SI of the
CMB polarization fields as well as the CMB foregrounds.